There are no conflicts in the placement of the data points
in the several graphical plots except for Figures Seven, Eight, and Ten, and
their requirements of placement in Figure Eleven. The designer could
have played with the numbers to provide us with the patterns shown.

We do not have evidence on the Flat and the Bent. Hence we
cannot make further estimates based on their construction.

The Data Display

The mathematical expressions, and the constants and
variables used in the pyramid designs, are all typical of what we find in
modern science and engineering. I need only open the pages of the
Handbook of Chemistry and Physics from the Chemical Rubber Publishing Company
to see them staring me in the face. Either the ancient designer/builder of the
Great Pyramids lived in a similar technological environment, or he anticipated
our environment. Either choice is unacceptable to modern minds.

Some may object that we are interpreting data according to
modern understanding, while the ancient designer may have performed according
to a completely different conceptual framework. This suggestion is refuted by
the exterior designs showing Pythagorean triangles, and the half-angle design
of the passages, all of which follow current mathematical understanding.
Further discussion will show the inadequacy of this suggestion.

To summarize:

1. We see the neat manner in which the designer provided
numerical values that would be easily discernible to an investigator of his
designs. These are 1/2Pi, 10 Pi, 10/2Pi, ln 10, and so on. This shows
his familiarity with mathematical and physical constants as we understand them
today.

2. He knew the difference between the common base (10) and
natural base (?) number systems. Of course one can be exchanged for the
other. Log x = ln x / ln 10, or ln x = log x / log ?. We can plausibly
infer that he understood logarithmic systems to any base. Certainly, if he
were familiar with one he would be familiar with another. If one wished to
express the regression lines on the graphs in the alternate system one could
easily do so. I merely left the one case in a base 10 expression to show
the similarity of results.

3. He was intimately familiar with exponential and
logarithmic functions. The manner in which he related two exponential decay
curves to provide the nearly linear relationship between the two, as
illustrated in Figure Five, is an outstanding piece of cleverness.

4. The use of Pi and e are identical to that which we use
in modern fields of engineering and physics.

5. He derived equations based on the variable N. This
variable is found in such modern developments as mathematical expressions for
cyclical or harmonic phenomena, and in infinite mathematical series. His plan
demanded a sufficient number of structures to permit graphical display that
would expose his design around that variable.

This has profound implications. He had to be fully
expectant that his project would be brought to completion with the number of
structures he planned. As we saw from the analysis this demanded the
complete gamut from Meydum to the Giza 1 Case. He had to be sure the social
resources were available to produce such massive works. But he also had
to be sure that the management of such a vast enterprise would achieve its
goals to the refinement we saw. This either meant that he was personally
present during the entire program, or that he taught it to others who
possessed similar determined purpose. If the kings merely believed the
pyramids were each intended for their individual fame then they could be short
of the knowledge and reasoning behind the project. That each was
intended individually for their personal honor and fame is the view of
history. Ancient records, written centuries afterwards, support such
belief. But if the designer/builder had a motive hidden from the eyes of
the kings, they would not understand his plan or methods. This means they
could not have been part of the project  except that the designer/builder
depended upon their support for the social resources. This explains why
Seneferu would have three structures to his credit, and why Khafre would
claim a pyramid that was built before the pyramid assigned to his
father. He was left with the prospect of making claim to something not
originally assigned to him. Again, this is why modern Egyptologists would make
that chronological assignment after Khufu.

6. He created equations based on r / 2? as a radian
measure. The multiplier of 1,000 was necessary to obtain a practical graphical
display because the angular measure was so small. One minute of arc =
approximately 1/3437 of a radian. By providing slopes on his curves to this
mathematical expression he showed us his familiarity.

Thus he was intimately familiar with circle concepts as we
understand today. He expressed the relationship of angular distances with
radian measure. The mathematical expressions from Figure Eleven are sensible
only if he used that form.

7. He recognized the use of incremental values nesting
inside physical lengths. The evidence hints at concepts that approach modern
calculus. How much he knew of differential mathematics is not certain. He
clearly gave us an indication of ability to design to such criteria.

8. He understood the mathematical concept of mean values.
Perception of the relationships among side deviations can only be understood
when compared against the means.

9. He understood the concept of mathematical absolute
numbers. Figures Four and Five use absolute numbers to achieve their
results.

10. He used graphical design methods. The results of
this study can only be understood if he drew out those relationships through
graphical analyses. Simple mathematical calculations would not make the
relationships obvious, nor evident, to mental perception  for himself, and
for a later investigator.

11. He understood manipulation of design parameters to
achieve presentation objectives without violating mathematical degrees of
freedom. With seven known structures, four sides, deviations on those sides,
and orientation of the individual sides, he had great freedom in his
development of mathematical displays. Again, his true genius is evident
through such finesse.

Such level of mathematics was not known until modern times.

1. The first treatise on algebra was written by Diophantus
of Alexandria in the 3rd century AD. The term derives from the Arabic al-jabr
or literally the reunion of broken parts.'' It gained widespread use
through the title of a book Ilm al-jabr wal-mukabala  the science of
restoring what is missing and equating like with like  written by the
mathematician Abu Jafar Muhammad (active c.800-847). He introduced the writing
down of calculations in place of using an abacus. But graphical analysis of relationships between
variables was not available until Rene Descartes (1596-1650), published the
first treatise on the geometric representation of algebraic equations in 1637.
The ancient Greeks were highly developed in space perception and used figures
to represent geometrical relationships but did not analyze them algebraically.
They did some analytical work but did not make mathematical associations which
would permit graphical display of physical relationships as we know them
today.

2. Tables for the use of logarithms were first published in
1614 by John Napier (1550- 1617). They were motivated by the need to alleviate
tedious trigonometric calculations for ship positions in the burgeoning
maritime traffic of the sixteenth and seventeenth centuries. The tables also
were highly useful for calculating compound interest for the financiers of
that aggressive commercial age. A1though the idea of multiplying numbers
through addition of logarithms goes back to Archimedes' study on mathematical
series (287 B.C.), with contributions from medieval mathematicians, it was not
until the recognition of the ease of substituting addition for multiplication
through trigonometric relationships by Wittich (1584) and Clavius (1593) that
led to practical developments.

3. The amazing mathematical power of the natural base
number e = 2.71828 . . . was not recognized until the late sixteenth century.
Although Leonhard Euler (1707-1783) was the first to use the letter e to
represent the number, tables of logarithms to the base e were first published
by Speidell in 1619.

4. The relationships demonstrated in the several figures
are those of applied mathematics. Logarithms, exponential functions, natural
base e, and radian measure are useful to physicists and engineers. Natural
decay in electrical circuits, mechanical systems, and physical processes
follow exponential forms. The solution of differential equations leads to both
common and natural base number representations.

5. Our knowledge of ancient Egyptian mathematics was
limited to two documents, the Moscow Papyrus and the Rhind Mathematical
Papyrus (6), both dating around 1900 B.C. They show only primitive mathematics
with elementary operations in arithmetic, calculation of areas and volumes,
and simple progressions. They did not use decimal multiplication and division;
they used a procedure of doubling of numbers to find multipliers and divisors
 nearly the same technique as employed in modern computers. Although this
procedure is useful for small numbers it rapidly becomes cumbersome for large
numbers.

6. The only other information we have on mathematical
knowledge of ancient times is found on the clay tablets of the Old Babylonian
empire, circa 1900 to 1600 B.C. Neugebauer and Sachs (7) published translation
and analysis which showed more advanced mathematical techniques than displayed
in the Moscow and Rhind Papyri. Scholars of Old Babylon used relatively
sophisticated algebraic expressions for geometric figures including circles,
segments of circles, diagonals of squares, right-angled triangles, (with
Pythagorean solutions), irregular triangles, trapezoids, irregular solids and,
most importantly, exponents and logarithms. Problems for exponents included
powers between 2 and 10. Problems for logarithms were given to base 2 and 16.

In all examples known to us the Babylonians illustrated
sophisticated problem solutions but just short of achieving theoretical
expressions. We have no evidence of their mathematical evolution or how they
achieved their knowledge.

7. The Egyptian papyri and the Babylonian clay tablets have
an important feature which has not been voiced. They both show learning by
rote. The methods of problem illustration are how to. This is the way
you multiply numbers; this is the way you calculate areas; and so on. Nowhere
do we have evidence of how the formulae were developed or the theories
deduced. As a consequence we believe they arrived at those methods through
trial and error, that there was no theoretical development. For this reason
many of us are unwilling to accept the more sophisticated examples for their
obvious sophistication. The solution of a truncated pyramid in the Moscow
papyrus is regarded with skepticism, although many such examples are found on
the Babylonian clay tablets. Then again the use of the Pythagorean
relationship 1500 years before Pythagorus for complex Pythagorean solutions is
received with even more difficulty. The evidence is contrary to our notions of
the slow evolution of mathematical knowledge. The real problem is that
advanced mathematics was known in the far distant past, and incorporated in
the pyramid designs, but then was lost to the world. Such phenomenon
could occur only if the society which possessed such knowledge was also dying
out.

8. If it took better than 400 or a 1,000 or 2,000 years to
develop modern mathematics how many centuries did it take ancient Egyptians,
members of the most conservative society known in history, to develop such
mathematics? Why is there no other record of it? If it is the product of a
technical culture where is the evidence for such culture? And why was it lost?

How did the designer know all this sophisticated
mathematics without a social base? How could he have developed it
entirely on his own, when we took millennia to achieve the same results?
We cannot reconcile such sophisticated knowledge without a social base for its
nurture  which then became lost to the world.